User blog:Holomanga/The Singularity in the Resourcespace Model
This blog post is available in .pdf form at . In Resourcespace, I derived the bound on the singularitarian resource generator density as \rho_\mathrm{h} \ll 10^{-9} \,\text{y}^\text{-1}\,\text{MW}^\text{-1} . In deriving this bound, I neglected the R^2 term resulting from the intelligence bulge. By including this term, some information about the dynamics of a technological singularity can be determined. Including the intelligence augmentation term gives the differential equation for a civilisation that has explored out to a radius R in resourcespace as \dot R = 2 \rho_\mathrm{e} R + \pi \rho_\mathrm{h} R^2 (0). If, at time t = t_0 , the civilisation has access to resources R = R_0 , this differential equation has an analytic solution of \frac{R}{R_0} = \frac{e^\alpha}{1 + \beta\left(1 - e^\alpha \right)} (1) with \alpha = 2 \rho_\mathrm{e} \left( t - t_0 \right) and \beta = \frac{\pi \rho_\mathrm{h} R_0}{2 \rho_\mathrm{e}} (2). The singularity occurs when R = \infty , so the denominator of equation (1) is zero, which means that \beta \left( e^\alpha - 1 \right) = 1 The time t_\mathrm{s} at which this occurs is t_\mathrm{s} - t_0 = \frac{1}{2 \rho_\mathrm{e}} \ln \left(1 + \frac{2 \rho_\mathrm{e}}{\pi \rho_\mathrm{h} R_0 }\right) . Limits of β The dynamics of the singularity are mostly controlled by the singularity parameter \beta , defined in equation (2), which is the ratio of the contributions from exponential growth to hyperbolic growth. When \beta is small, equation (1) becomes (to second order in e^\alpha ) \frac{R}{R_0} = e^\alpha \left(1 - \beta\right) + \beta e^{2\alpha} which is simply exponential growth e^\alpha , plus a small perturbation from intelligence that is proportional to \beta and vanishes when \beta \ll 1 . When \beta is large, equation (1) becomes \frac{R}{R_0} = \frac{e^\alpha}{\beta\left(1 - e^\alpha \right)} . If the singularity parameter is large, then we know from equation (2) that \rho_\mathrm{e} is small (this also assumes that t \approx t_0 \approx t_\mathrm{s} ; it is always possible to find a time far enough before the singularity such that the contribution from intelligence augmentation is small), so \alpha is also small, so expanding e^\alpha to first order in \alpha gives \frac{R}{R_0} = -\frac{1}{\beta \alpha} - \frac{1}{\beta} . This is simply the hyperbolic growth that one would expect in the absence of the primitive line, plus a small extra term -\frac{1}{\beta} that vanishes when \beta \gg 1 . Limits of α Very long before the singularity, \alpha is large and negative, so e^\alpha \ll 1 . This means that equation (1) reduces to \frac{R}{R_0} = \frac{e^\alpha}{1 + \beta} , which is just simple exponential growth with a correction factor \frac{1}{1 + \beta} to remove the effects of the upcoming singularity on R_0 . At a small time (much less than one exponential doubling time) \delta t before the singularity, t = t_0 + \frac{1}{2 \rho_\mathrm{e}} \ln \left(1 + \frac{1}{\beta} \right) - \delta t , so \alpha = \ln \left( 1 + \frac{1}{\beta} \right) - \delta t . So e^\alpha = e^{-2 \rho_\mathrm{e} \delta t} \left( 1 + \frac{1}{\beta} \right) . Since \delta t is small compared to \rho_\mathrm{e} , e^\alpha = \left( 1 - 2 \rho_\mathrm{e} \delta t\right) \left( 1 + \frac{1}{\beta} \right) . Substituting this into (1) and ignoring higher-order terms in \rho_\mathrm{e} \delta t gives R = \frac{1}{\pi \rho_\mathrm{h} \delta t} . This is just hyperbolic growth that reaches infinity at the singularity, as expected. Interestingly, the amount of resources actually increases when \rho_\mathrm{h} is small. This is because, for small \rho_\mathrm{h} , more resources are needed for the singularity to begin, so at a small time before the singularity more resources will be present. Since this depends only on \rho_\mathrm{h} and \delta_t , all singularities look the same on a log-plot sufficiently close to the singularity (taking the same amount of time to go from a doubling time of four years to a doubling time of one year, for example). These effects are prominent in figures 1, 2, 3 and 4. Predictions With an analytic solution available, it is possible to use measured data in order to make some predictions about the singularity. For human civilisation, R_0 = 21\times10^6 \,\text{MW} , t_0 = 2019 \,\text{AD} , and \rho_\mathrm{e} \approx 10^{-2} \,\text{y}^\text{-1} . Solving this for the most optimistic singularitarian growth rate, \rho_\mathrm{h} \approx 10^{-9} \,\text{y}^\text{-1}\,\text{MW}^\text{-1} , gives a technological singularity at around 2031, shown in figure 1. This optimistic view seems to be clearly wrong: it predicts an economic growth rate of 8.6%, rather than the ~2% that is actually occuring. Unless some clear signs of the singularity happen soon (GPT-2, anyone?), this is evidence that \rho_{\mathrm{h}} is much smaller than this. Part of this effect is that some of the economic growth that is attributed to industry is actually from the early signs of the singularitarian mode, so some double-counting is occurring, though even if we assume all the growth we see today is from the singularity (unlikely - factories still matter), there is still a 6.6% discrepancy to be explained. This optimistic view also predicts that the signs of the singularity will become visible very soon: the first doubling of the economy occurs by 2025 (6 years), the second doubling by 2028 (3 years), the third doubling by 2030 (2 years), and all subsequent doublings during 2031 (< 1 year). In fact, in this scenario, the technological singularity occurs before the next doubling of the economy due to industry. A situation more in line with what we observe would occur for a smaller value of \rho_\mathrm{h} . If we assume that the singularity currently accounts for no more than 60% of current economic growth (via identifying the very computerised service sector with the singularity; obviously this is absurdly optimistic), then \rho_\mathrm{h} \approx 10^{-10} \,\text{y}^\text{-1}\,\text{MW}^\text{-1} . This value for \rho_\mathrm{h} gives a predicted singularity in 2088 and is plotted in figure 2. If we assume that no measurable economic growth is due to the singularity, then this means that the current contribution to the growth rate from hyperbolic growth occurs is smaller than 0.05%. This occurs when \rho_\mathrm{h} \approx 10^{-11} \,\text{y}^\text{-1}\,\text{MW}^\text{-1} , predicts a singularity occurring in the year 2190, and is plotted in figure 3. Finally, the most pessimistic account still consistent with my model of civilisation growth would have the singularity happen at around the time the civilisation becomes a type 2 civilisation. This occurs when \rho_\mathrm{h} \approx 10^{-21} \,\text{y}^\text{-1}\,\text{MW}^\text{-1} , and gives a singularity occurring in the early 4th millennium. A plot of this is shown in figure 4. Further Development One way this model could be expanded is by adding additional terms to capture higher-order effects. If both \rho_\mathrm{e} and \rho_\mathrm{h} are small, then a polynomial term \propto R^\frac{2}{3} would dominate (a space-opera like scenario with a slow expansion through space with close to modern technology). Along these lines, in reality there are multiple exponential growth modes (forager, farmer, industrial), each with their own associated growth rates. There may be some way to modify the differential equation to capture this sort of behaviour. Adding an additional exponential growth mode (the age of Em) would give a brief period of accelerated exponential growth before the singularity proper occurs. This would change the dynamics of the singularity somewhat (for example, there could be a singularity happening soon but with a \rho_\mathrm{h} that is very low), and a quantitative account of this would be likely to give a lot of insight. The model is clearly flawed for the singularitarian regime, because, in reality, R does not go to \infty , and saturates at some maximum. This could be resolved by modifying the R^2 term to capture this behaviour, such as by replacing it with term that saturates like R^2 \left( 1 - \frac{R}{R_\mathrm{s}} \right) (this term is flawed, because it predicts strong negative growth for R > R_\mathrm{s} , but could be studied in isolation or with some modificaton). If this addition is made, then it opens up the possibility of adding multiple hyperbolic growth modes that saturate at different points. Such a pattern of growth appears in the fictional universe of Orion's Arm, where there are multiple singularity levels, which, in this framing, represent the endpoints of different regimes of hyperbolic growth. One key uncertainty in this model is how much current growth is driven by the exponential, rather than the hyperbolic, growth modes. One way to determine this might be to fit a model corresponding to equation (0) to recent economic growth data to get bounds on the coefficients. Some further qualitative research may be valuable, too. In the lead-up to the singularity in this model, the major impacts of the singularity occur over a period of around thirty years, which is sufficient time for major social changes to happen. An accounting of what these social changes might look like (in various scenarios) would be very interesting. In addition, a smaller \rho_\mathrm{h} leads to a longer period of time prior to the singularity at which economic growth is at industrial levels. The social consequences of this constant growth with no qualitative changes over different periods of time, and what this entails for the starting conditions as the singularity is entered, would be interesting. For example, for large \rho_\mathrm{h} , AGI is likely to be achievable with modern technology and few insights, and hence be prosaic. Category:Blog posts